Theorems · Theorem · probability
MeasureTheory.submartingale_nat
∀ {Ω : Type u_1} {E : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : NormedAddCommGroup E]
[inst_1 : NormedSpace ℝ E] {𝒢 : MeasureTheory.Filtration ℕ m0} [CompleteSpace E] [inst_3 : PartialOrder E]
[IsOrderedAddMonoid E] [ClosedIciTopology E] [IsOrderedModule ℝ E] [MeasureTheory.IsFiniteMeasure μ] {f : ℕ → Ω → E},
MeasureTheory.StronglyAdapted 𝒢 f →
(∀ (i : ℕ), MeasureTheory.Integrable (f i) μ) →
(∀ (i : ℕ), f i ≤ᵐ[μ] μ[f (i + 1) | ↑𝒢 i]) → MeasureTheory.Submartingale f 𝒢 μ- Defined in
- Mathlib.Probability.Martingale.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 300 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites30
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- MeasurableSpacestatement and proof · cited by 13,106
- NormedSpacestatement and proof · cited by 12,499
- MeasureTheory.Measurestatement and proof · cited by 10,939
- PartialOrderstatement and proof · cited by 6,410
- CompleteSpacestatement and proof · cited by 2,532
- MeasureTheory.aestatement and proof · cited by 2,352
- le_reflproof · cited by 2,061
- Filter.univ_mem'proof · cited by 1,672
- IsOrderedAddMonoidstatement and proof · cited by 1,659
- Filter.mp_memproof · cited by 1,537
Cited by2
Results whose statement or proof uses this declaration.
- MeasureTheory.submartingale_of_condExp_sub_nonneg_natproof · cited by 2
- MeasureTheory.supermartingale_natproof · cited by 0